Chaotic linear maps

Nonlinear Analysis 45 (2001) 707–721www.elsevier.nl/locate/na

Chaotic linear maps

Stefano Lencia, Renzo Lupinib; ∗

aIstituto di Scienza e Tecnica delle Costruzioni, Universit�a di Ancona, via Brecce Bianche,Monte d’Ago, 60131 Ancona, Italy

bDipartimento di Matematica, Universit�a di Ancona, via Brecce Bianche, Monte d’Ago,60131 Ancona, Italy

Received 6 April 1998; accepted 14 September 1999

Keywords: Chaos; Linear maps; In-nite-dimensional spaces; Shift map

1. Introduction

In dynamical systems theory the fundamental objective is to characterize the limit setsof sequences generated by recurrence laws or maps (Poincar3e maps for systems withcontinuous time). For -nite-dimensional space states the asymptotic properties of se-quences do not depend on the metric used; in such spaces bifurcations can only occur infamilies of maps. Instead, in in-nite-dimensional spaces, because of the non-equivalenceof the metrics, we can have metric-dependent asymptotic properties for the same map,that is bifurcations in families of metrics.

Beside this, the substantial changes in the structure of the spectrum of linear mapsinduced by the transition from -nite- to in-nite-dimensional spaces can be given inter-esting interpretations within the context of the dynamical systems theory, in particularas regards to the central notion of chaoticity.

Let us recall that in -nite-dimensional spaces nonlinearity is a necessary conditionfor chaoticity. In fact, for linear maps one of the conditions for chaoticity [1], thedensity of periodic points, is incompatible with another one, the sensitivity to initialconditions. But, as we shall show in this work, this is no longer true for maps inin-nite-dimensional spaces, i.e. there exist linear, continuous maps in Banach spaceswhich exhibit both a dense set of periodic points and sensitivity to initial conditions.Of course non-compacteness of the map is necessary for this to be true; in fact density

∗ Corresponding author. Tel.: +390-71-583-485.

0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(99)00418 -6

708 S. Lenci, R. Lupini / Nonlinear Analysis 45 (2001) 707–721

of periodic points implies density of eigenvalues on the unit circle, whereas compactlinear maps have discrete spectrum. But as we pass from -nite- to in-nite-dimensionalspaces, compacteness is lost in most linear maps.

In the present paper, we aim at clarifying the statement made above concerning thecompatibility of linearity and chaoticity by taking into consideration a well-known mapthat, in fact, represents the prototype of all chaotic maps, the shift map.

As it is well known [1,5], such a map operating in a suitable metric space ofsequences (without the structure of vector space) is conjugate to the logistic map,a one-dimensional nonlinear map. Moreover, as we shall show in the present paper,we can conjugate any map in -nite dimension to the shift map in a suitable Banchspace of sequences where it turns out to be linear and continuous. We shall identifycompact subsets of such Banach space where the shift map is chaotic. Finally, weshall conjugate the shift map to other simple linear maps (involving translations anddiDerentiation) operating in Banach spaces of analytic functions, thus providing more“concrete” examples of chaotic, linear maps.

The following notations will be used henceforth. N is the set of natural numbers,Z is the set of integers, Q is the set of rationals, C is the set of complex numbers, Rand R+ are the sets of real numbers and the set of positive real numbers, respectively.Given sets X and Y then F(X; Y ) denotes the set of functions de-ned in X with rangein Y . The set of the s-uple’s of elements of X , usually denoted as X s, can be interpretedas the set of functions from {1; 2; : : : ; s} to X , that is F({1; 2; : : : ; s}; X ). Likewise, theset of sequences whose elements belong to X , often denoted as XN, will be interpretedas the set of functions f : N→ X and denoted as F(N; X ). Given n∈N; f(n) will becalled the nth component of f. The range of f is just the sequence of its components.

A function f∈F(X; X ) is also called map in X and the couple (f; X ) a (discrete-time) dynamical system. A subset E⊆X is called f-invariant if f(E) =E and f-trappingif f(E)⊆E. More generally, a sequence of subsets En; n∈N; is called f-invariant(f-trapping) if f(En) = En+1 (f(En)⊆En+1); n∈N. When the En’s are composed ofa single point they constitute a trajectory of f. When E is invariant and constitutedby p points then it is a periodic trajectory of period p.

2. The spaces F�(N;X)

Let X be a -nite-dimensional, normed space [3,4] with C as the set of scalars (e.g.X=Cs; s a natural number). The elements of X will be denoted by bold latin letters.With the usual linear operations in F(N;X),

(f + g)(n) = f(n) + g(n);

( f)(n) = (f(n));

(i.e. “component by component”) F(N;X) becomes a vector space with zero the func-tion o(n) = 0; ∀n∈N; 0 being the null vector in X. If X is an algebra with unit e(like the set of matrices s× s), then F(N;X) becomes an algebra with respect to theusual product

(fg)(n) = f(n)g(n)

S. Lenci, R. Lupini / Nonlinear Analysis 45 (2001) 707–721 709

with unit u(·) such that u(n) = e; ∀n∈N. We can also de-ne the exterior multiplica-tion of elements of F(N;C) by elements of F(N;X) by

(’f)(n) = ’(n)f(n);

where ’(n)∈F(N;C) and f∈F(N;X). In the same way, the exterior multiplicationof elements of F(N;C) by elements of X is de-ned by

(’x)(n) = ’(n)x;

where ’(·)∈F(N;C) and x∈X. As regards to the algebraic properties it is clear thatF(N;C) behaves as a space of scalars for F(N;X) and for X. The following additionalde-nitions and notations for operations on sequences will be useful.

1. For natural n and m; n ≤ m, the (n; m)-segment of f∈F(N;X), denoted asRn;m(f); is the restriction of f to the segment {n; n + 1; : : : ; m} of N (the range ofRn;m(f) is the (n; m)-segment of the range of f). It is also allowed for m to be ∞.For short we also write Rn = R1; n. Rn(f)∈F({1; : : : ; n};X).

2. The inverse of segmentation is the concatenation of functions. Giveng∈F({1; : : : ; m};X), and f∈F(N;X) the concatenation of g and f is de-ned as({g; f})(n) = g(n) if n ≤ m, and ({g; f})(n) = f(n − m), if n¿m. In particular,we shall de-ne the projection of f∈F(N;X) on the -rst segment of length n asSn(f) = {Rn(f); o(·)}, whose range is the sequence {f(1); f(2); : : : ; f(n); 0; 0; : : :}.

3. By periodic repetition of g∈F({1; 2; : : : ; s};X) we mean the periodic function ofF(N;X) given by P(g)(n) = g([n]s), being [n]s = n (mod s). Explicitely, the range ofP(g) is the periodic repetition of the range of g, i.e.

{g(1); g(2); : : : ; g(n); g(1); g(2); : : : ; g(n); g(1); g(2); : : : ; g(n); : : : :}:In particular, the range of P(Rn;m(f)), also denoted as Pn;m(f), is the periodic repetitionof the segment of components of f between the nth and the mth. Moreover, Pn =P1; n

denotes the periodic repetition of the -rst n-segment.4. Exponential functions in F(N;C) are de-ned as follows. Given ∈C the ex-

ponential function of base , that is n → n, is denoted as (·). In particular, givenx∈X; 1(·)x is the constant function f(n) = x.

In this paper, a basic role is played by the following class of subspaces of F(N;X).For any �∈R+, we de-ne

F�(N;X) ={f: f∈F(N;X); sup

n∈N{|f(n)|�n}¡ + ∞

};

where | · | denotes the norm in X. It is clear that F�(N;X) is a linear subspace ofF(N;X) and that |f|� = supn∈N{|f(n)|�n} is a norm in it. Moreover, from

�n|f(n)| ≤ |f|�;∀n; it follows that the projection unto the nth component f→f(n) is a continuos linearfunction from F�(N;X) to X. In particular, if a sequence fi; i∈N, is convergent inF�(N;X) (norm-convergent) to f, then fi is also convergent to f component bycomponent in X, that is limi→∞ fi(n) = f(n); ∀n∈N (weak convergence). Likewise,if fi is Cauchy in F�(N;X) then it is also Cauchy “component by component”, sothat it is also weakly convergent.


Theorem 1. With the norm | · |�; F�(N;X) is a non-separable Banach space.

Proof.. To prove completeness of F�(N;X), let us consider a Cauchy sequence fi; i∈N.As remarked above for any given n; fi(n) has a limit as i→∞, say xn ∈X. By Cauchycondition, ∀�¿ 0 there exists k� such that ∀i; j ¿ k�

�n|fi(n) − fj(n)| ≤ |fi − fj|� ¡ �:

Taking the limit as j tends to in-nity we -nd that, for every n∈N and i¿ k�

�n|fi(n) − xn|¡�

which implies that f; de-ned by f(n) = xn; belongs to F�(N;X) andlimi→∞ |fi − f|� = 0.

Let us now show that F�(N;X) is non-separable. Let e∈X be of unitary norm andlet us consider the subset of the functions of the form g(n)= �−nun, where un ∈{0; e}.Formally, g(·)∈ (�−1)(·)F(N; {0; e}). Clearly, this set is not countable and belongs toF�(N;X), as |g|� ≤ 1. The distance between two distinct g’s is clearly 1. Let us considerthe set of disjoint balls of positive radius �¡ 1

2 centered at the g’s. Any dense set hasto have at least one point in each of these balls, so that it is not countable.

We now remark some properties of the spaces F�(N;X).1. F1(N;X) = l∞, being l∞ is the common notation for the space of bounded

sequences in X, and |f|1 is the sup- or uniform norm (usually denoted as | · |∞).F1(N;X) is also a Banach algebra. Note that |f|� = |(�−1)(·)f(·)|1.

2. As a function of � the F�(N;X) constitute a nested family of spaces. For �¡�;| · |� ¡ | · |�, that is F�(N;X) is continuously embedded in F�(N;X). In fact, the char-acteristic condition for functions in F�(N;X) is to be of exponential growth boundedby −ln �.

3. For 0¡�¡ 1; l∞ is a subset of F�(N;X), but it is not dense in F�(N;X). Infact, let x∈X and a positive number �¡ |x| be given. Let us assume that there existg∈ l∞ such that |(�−1

)(·)x− g(·)|� ¡ �. Then

|(�−1)(·)x− g(·)|� = sup

n∈N{|x− g(n)�n|} ≤ �

implying |g(n)| ≥ (|x| − �)�−n, which contradicts g∈ l∞.4. For every �∈R+; �¡�, and x∈X, the function (�−1)(·)x belongs to l∞,

the closure of l∞ with respect to the �-norm. In fact, the sequence of projectionsSn((�−1)(·)x), having ranges {�−1; �−2; : : : ; �−n; 0; 0; : : :}x, belongs to l∞ andconverges to (�−1)(·)x, as

|(�−1)(·)x− Sn((�−1)(·)x)|� ≤ supk¿n

{(�=�)k}|x| = (�=�)n|x|:

5. For 0¡�¡ 1; limn→∞ Sn(f) = f, and limn→∞ Pn(f) = f; ∀f∈ l∞, as

|Sn(f) − f|� =(

supk ∈N

|f(n + k)|�k)

�n ≤ �(

supk ∈N

|f(k)|)

�n = |f|1�n+1


and, in a similar way,

|Pn(f) − f|� ≤ 2|f|1�n+1:

6. For 0¡�¡ 1; f and g∈ l∞ and such that Sn(f) = Sn(g) (n given)

|f − g|� ≤ �n+1|f − g|1:The last two properties play a fundamental role in all the analysis to follow. In fact,

from now on we shall assume, unless otherwise stated explicitely, 0¡�¡ 1.We introduce another class of subsets of F�(N;X). Let Bn; n∈N, be a sequence

of compact subsets of a given compact subset B of X. Let us denote by F(N; {Bn;n∈N}) the set of functions such that f(n)∈Bn; ∀n∈N. It is clear that such a setbelongs lo l∞, so that (because of 0¡�¡ 1) it also belongs to F�(N;X). Thereforeit will be also denoted F�(N; {Bn; n∈N}), and F�(N; B) when Bn = B; ∀n, in orderto point out that the �-norm is de-ned in it.

From the previous remarks it follows that ∀f∈F�(N; B) (and 0¡�¡ 1) the fol-lowing inequality holds:

|Pn(f) − f|� ≤ �(B)�n+1;

where �(B) denotes the diameter of B.

Theorem 2. If 0¡�¡ 1; Bn compact in X; Bn ⊆B; ∀n; B compact; then F(N; {Bn;n∈N}) is compact in F�(N;X).

Proof.. When B is a single element the theorem is trivial. Let us then assume that thediameter �(B) of B is positive. Let us show -rst that F(N; {Bn; n∈N}) is closed. Letfi be a sequence in F(N; {Bn; n∈N}) converging to f, and let us show that f belongsto F(N; {Bn; n∈N}. In fact, because of weak convergence of fi to f (i.e. componentby component), it follows that, for any given n; fi(n) converges to f(n)∈Bn, as Bn

is closed. Therefore f∈F(N; {Bn; n∈N}). In conclusion, F(N; {Bn; n∈N}) and, inparticular, F(N; B); are closed sets.

To complete the proof it is suJcient to show that F(N; B) is compact. As this setis closed, it is suJcient to show that ∀�¿ 0 there exists a -nite �-net in F(N; B). Bbeing compact there exists a -nite �-net in B, say !̃ = {x1; x2; : : : ; xm}; that is, ∀x∈Bthere exists k; 1 ≤ k ≤ m, for which |x − xk |¡�. If needed, we can add points tothe �-net !̃ in such a way that �m ¡�. For any combination, say ", of {1; 2; : : : ; m}let us denote as f" the function of F(N; B) whose range is the periodic repetitionof the �-net "(!̃), that is f"(n) = x"([n]m), being [n]m = n (modm). Let us call R� theset of such functions, and let us show that it constitutes a -nite �-net for F(N; B).Clearly, R� is a subset of F(N; B) that contains no more than mm elements. Moreover,let f∈F(N; B). Then, as !̃ is a �-net for B, there exists a permutation, say "∗, suchthat |f(k) − x"∗(k)|¡� for 0 ≤ k ≤ m. Therefore,

|f"∗ − f|� ≤ sup

{sup

1≤k≤m{|f(k) − x"∗(k)|}�k ; �(B); �m+1

}

≤ �� sup{1; �(B)}:


3. The shift map # in F(N;X)

In F(N;X) the shift map [2,6,7] denoted by #, is de-ned by shifting by one theargument of the functions, that is

(#(f))(n) = f(n + 1):

# is well de-ned in F(N;X), but we shall consider only its restriction to a givensubspace F�(N;X), which is clearly #-invariant. The basic properties of # as a mapin F�(N;X) are now discussed.

Theorem 3. The shift map # :F�(N;X) → F�(N;X) is (i) surjective; (ii) linear;(iii) continuous; (iv) non-compact and (v) not invertible. Furthermore; its normis �−1.

Proof. (i), (ii) and (v) are obvious. (iii) follows from the inequality |#(f)|�≤ �−1|f|�. Next, let us consider the element of F�(N;X) (�−1)(·)u, where u is a unitaryelement of X. We have |(�−1)(·)u|� =1 and |#((�−1)(·)u)|� =�−1, which proves the lastassertion. To prove (iv) let us consider the sequence of functions gk ; k ∈N, de-nedas gk(n) = �−n�k;nu (�k;n is the Kroenecker delta). This sequence in F�(N;X) isbounded, as |gk |� =1, but #(gk) has no Cauchy subsequence, as |#(gk)−#(gj)|� = �−1

∀k; j; k �= j.

Clearly, # has ∞s inverses in F�(N;X). Let us discuss only the case X = C. Wecan associate with # the in-nite matrix of scalars, say A, whose element (i; j) is �j; i+1.Using multiplication “rows by columns”, we have the following matrix representationof #:

#(f)(N) = Af(N); (1)

in terms of function ranges. Since AAT = I and A{1; 0; 0; : : : ; 0; : : :} = {0; 0; : : : ; 0; : : :},where I is the in-nity identity matrix, Eq. (1) can be given the inverse form f(N) =ATf(N) + {1; 0; 0; : : : ; 0; : : :}; being an arbitrary complex number.

As regards to the spectrum of # in F�(N;X), it is easy to see that 0 is one ofthe eigenvalues of #, the associated eigenspace being the set of functions such thatf(1) = x and f(n) = 0 for n¿ 1, or {x; o(·)}. Another obvious eigenvalue is 1, witheigenspace constituted by the constant functions, i.e. 1(·)x. These are particular casesof the following theorem.

Theorem 4. The point-wise spectrum of # is the disk of C of equation |�| ≤ �−1 andthe essential spectrum is empty. The eigenspace associated with each eigenvalue � isthe s-dimensional space �(·)X.

Proof. The equation #(f) = �f, namely f(n+1) = �f(n), is solved by f(n) = �n−1f(1),that is f(·) = �(·)x; x∈X. However, �(·)x∈F�(N;X) if and only if |�| ≤ �−1. The factthat the essential spectrum is empty is a consequence of |#|� = �−1, as demonstratedabove.


We shall consider now the couple (#; F�(N;X)) as a (∞-dimensional) dynamicalsystem. Such a map “synthesizes” many s-dimensional dynamical systems in X, asexplained in the next theorem, for which the following de-nitions and notations areneeded.

First we recall that, the map f1 ∈F(X1; X1) is conjugated to the map f2 ∈F(X2; X2)in a trapping or invariant subset E1 ⊆X1 if there exists a homeomorphism of E1

unto a subset of X2 such that (f1(x)) = f2( (x)); ∀x∈X1, namely, ◦f1 = f2 ◦ .Then let f ∈F(X;X) be a (s-dimensional) map in X, and let B be a f-trapping

subset of X. Given x∈B we denote as fNx the function of F(N;X) de-ned as fNx (n)=f (n)(x). Equivalently, fNx is such that its range is the trajectory of f through x, that is{x; f(x); f (2)(x); : : : ; f (n)(x); : : :}. We denote as Of the correspondence x→ fNx , from Bto F(N;X) generated by f .

Theorem 5. Let f ∈F(B;X) be Lipschitz-continuous in B⊆X; B being f-trapping.Then there exists �∗ ∈ (0; 1) such that f in B is conjugated to the shift map # in atrapping subset of F�∗(N; B).

Proof.. Let us show that for a suitable value of � the function Of , de-ned above,is a homeomorphism between B and and its image in F�(N; B). In fact, f beingLipschitz-continuous in B there exists M ∈R+ such that ∀x; y∈B; |f(x) − f(y)| ≤M |x− y|. We have

|Of(x) − Of(y)|� = |fNx − fNy |�= sup

n≥0|f (n)(x) − f (n)(y)|�n+1 ≤ sup

n≥0(M�)n�|x− y|

and

|Of(x) − Of(y)|� = |fNx − fNy |�= sup

n≥0|f (n)(x) − f (n)(y)|�n+1 ≥ �|x− y|:

Therefore if we choose �∗ ¡M−1, then Of maps homeomorphically B into F�∗(N; B).Finally, #(Of(x))(n) = #(fNx )(n) = f (n+1)(x) = f (n)(f(x)) = fNf(x)(n) = Of(f(x))(n);∀n∈N, implies # ◦ Of = Of ◦ f .

Remark 1. When B is compact, f-trapping (f-invariant), and f is continuouslydiDerentiable in B, then f is also Lipschitz-continuous in B and the conclusion ofTheorem 5 holds. In such a case F�∗(N; B) is compact and Of(B) is a compact#-trapping (#-invariant) subset of it.

Remark 2. Theorem 5 can be easily extended to f-permanent chains {B0; B1; B2; : : : ;Bn; : : :} of subsets of X.

Remark 3. Regarding the viceversa of Theorem 5, we can reason as follows. Let Pbe a #-trapping subset of F�(N;X) satisfying the following: there exists M ∈R such


that |f(n) − g(n)|¡M�−n|f(1) − g(1)|; ∀f; g∈P. Let E1 = {f(1); f∈P}. Let usconsider the mapping, say O, from P to X, de-ned as O(f) = f(1). By assumptionon P; O is an homeomorphism from P to E1. Next, we de-ne the map f from E1

to X by f(f(1)) = f(2). By construction, f is Lipschitz-continuous and O(#(f)) =f(2) = f(f(1)) = f(O(f)); ∀f∈P. This shows that f and # are conjugated throughthe mapping O.

Other obvious invariant subsets of # are its eigenspaces, but the dynamics restrictedto the eigenspaces associated with |�| �= 1 is trivial. In fact, ∀x∈X,limn→∞ #n(�(·)x) = (limn→∞ �n)�(·)x = 0, if |�|¡ 1, while limn→∞ |#n(�(·)x)| = ∞,if |�|¿ 1. Let us denote as C1 the unit circle in the complex plane. We call stationarythe unitary eigenvalues and their eigenvectors. We also call rational or irrational astationary eigenvalue �=e2"i#; #∈ [0; 1] and the corresponding eigenvectors dependingon being # rational or irrational.

We de-ne the subsets of F�(N;C) K = {(e2"i#)(·); #∈ [0; 1]} and Kr = {(e2"i#)(·);#∈Q ∩ [0; 1]}. Moreover, given p∈N, we denote as +k = e2"i(k=p); k = 0; 1; 2; 3; : : : ;p − 1, the set of distinct p-roots of unity, and the subset of F�(N;C) Kp = {+(·)

k ;k = 0; 1; 2; : : : ; p− 1}. Clearly Kr =

⋃∞p=1 Kp.

K; Kr and Kp are sets of stationary eigenvectors for # in F�(N;C). Correspondingsets of stationary eigenvectors for # in F�(N;X) can be represented as KE; KrE andKpE where E is a subset of X.

We interpret K as the image in F�(N;C) of C1 according to the function � → �(·)

from C to F�(N;C). This function is continuous. In fact, let �; + belong to C1. From�n −+n =(�−+)(�n−1 + �n−2++ �n−3+2 + · · ·++n−1) we deduce |�n −+n| ≤ |�−+|n.Thus, |�(·) − +(·)|� ≤ |�− +| supn∈N{(n− 1)�n}.

From continuity of � → �(·) and compactness of C1 it follows that K is compactand that Kr is dense in K .

Remark 4. Given ’(·)∈F�(N;C) and x∈X we have #(’(·)x) = #(’(·))x. Thereforethe dynamics of # in (invariant) subsets of the form PE; P⊆F�(N;C) and E ∈X, isequivalent to the dynamics of # in P. In particular, it is not restrictive when studyingthe dynamics of # in subsets of eigenvectors to assume X= C.

Theorem 6. The set of periodic points of # in F�(N;C) is the linear subspace ofF�(N;C) generated by Kp.

Proof.. In fact, the equation #p(f) = f implies that f is an eigenvector of # corre-sponding to a pth root of unity. Viceversa, from +1+p

k = +k , it follows #p(+(·)k ) = +(·)

k ,which means that +(·)

k is a p-periodic point. On the other hand, the set of the p sta-tionary eigenvectors +(·)

k ; k = 0; 1; 2; : : : ; p, is linearly independent because the matrixof the -rst p-segments of length p of the +(·)

k ’s is the (non-singular) Vandermondematrix generated by the +k ’s.


(The set of p-periodic points of # in F�(N;X) is the linear space generated byKp{u1; u2; : : : ; us}, where {u1; u2; : : : ; us} is a base in X.)

If � is an irrational eigenvalue then �n �= � for every n∈N, so that trajectoriesstarting from irrational eigenvectors are not periodic. The set K is not invariant, butusing the eigenspaces we can construct another invariant subset of #, that is H ={cf: |c|=1; f∈K}=C1K . It will be called the stationary set. Along with H we shallconsider also Hr ={cf: |c|=1; f∈Kr}=C1Kr . Clearly Hrat, which is made of periodicpoints, is dense in H . Moreover, H is a compact subset of F(N; C1), which is itselfcompact and invariant.

4. Chaoticity of #

Let us recall the relevant de-nitions pertaining to the concept of chaotic map[1,2,5–7]. Let it be given that a map f :X → X; X a metric space, and a compactsubset R⊆X , is invariant under f.

The following de=nitions formalize the concept of chaotic dynamics.1. f is said to have sensitivity to initial condition (henceforth SIC) in R if there

exists �¿ 0 such that, for any x∈R and any neighborhood U of x, there exists y∈Uand n¿ 0 such that d(f(n)(x); f(n)(y))¿�.

2. R is said to be (topologically) transitive if there exists y∈R such that thetrajectory through y is dense in R, namely, for any given x∈R and �¿ 0, there existssome integer n such that d(f(n)(y); x)¡�.

3. f is said to be chaotic in R if (i) f has SIC on R; (ii) R is topologicallytransitive and (iii) the set of periodic points of f is dense in R.

We remark that there is not a general consensus as regards to the conditions for chaosgiven above. For example some authors [7] do not require property (iii) and othersrelax the topological conditions for R [1]. The latter point is particularly importantfor in-nite-dimensional space states because of the non-equivalence of the metrics.Anyway, in the present paper we shall adopt the conditions for chaoticity listed above.

Theorem 7. Let B be a compact subset of X with more than one point. Then # ischaotic in F�(N; B).

Proof.. We have already remarked above the fact that F�(N; B) is both compact andinvariant. Let �(B) be the diameter of B. As B contains more than one point thereexists also a positive number, say d(B), such that given x∈B there exists y∈B, forwhich |x − y|¿d(B).

To demonstrate the density of periodic points it is suJcient to note that, givenf∈F�(N; B); Pn(f) is periodic and |Pn(f) − f|� ≤ �n+1�(B).

To demonstrate SIC let f∈F�(N; B) be given. There exists y∈B such that |f(n +1)−y|¿d(B). Consider g={Pn(f); 1(·)y}, that is g(k)=f(k) for k ≤ n, and g(k)=yfor k ¿n. Clearly g∈F�(N; B). Sensitivity follows from the inequalities |f − g|� ≤�n+1�(B), and |#n(f) − #n(g)|¿�d(B).


We recall that B, being compact, is separable. Let y∞ = {y1; y2; : : : ; yn; : : :} be asequence dense in B. We can obtain a new dense sequence in the following way. Wesubstitute in the above sequence y2 with the 22 combinations of {y1; y2}; y3 with the33 combinations of {y1; y2; y3}; yn with the nn combinations of {y1; y2; : : : ; yn}, and soon. Let us call �∞ = {�1; �2; : : : ; �n; : : :} this new sequence. Clearly, �∞ contains asa segment any combination of the previous sequence y∞. Let g∗ be the function ofF�(N; B) whose range is �∞. We show that the trajectory of # through g∗ is dense. Infact, let �¿ 0 and f∈F�(N; B) be given. Let n be such that �n ¡ � and let us considerthe range of the -rst n-segment of f, that is {f(1); f(2); : : : ; f(n)}. Because of thedensity of �∞ there exist a set of n elements in y∞, say {yk(1); yk(2); : : : ; yk(n)}, suchthat |f(i)− yk(i)|¡� for i=1; 2 : : : ; n. By construction {yk(1); yk(2); : : : ; yk(n)} occurs asa segment of �∞, that is there exists m such that �m+i = yk(i) for i = 1; 2 : : : ; n. Nowtransitivity follows from the inequality

|#m(g∗) − f|= sup

{sup

0≤i≤n|f(i) − yk(i)|�i; sup

i¿n|f(i) − g∗(m + i)|�i

}

≤ �� + �n�(B) ≤ �(1 + �(B)):

Remark 5. Theorem 7 can be generalized to subsets of the form F�(N; {B1; B2; : : : ;Bn; : : :}) where {B1; B2; : : : ; Bn; : : :} is a nested sequences of compact sets, that isBk ⊇Bk+1;∀k ∈N, and inf n �(Bn)¿ 0.

Remark 6. When B is a discrete subset of C or R, Theorem 7 reduces to a well-knownresult of Symbolic Dynamics, see [7].

Remark 7. If B⊆X is compact, f-invariant, and f is chaotic in B then, accordingto Theorem 5, # is chaotic in Of(B). The inverse problem, that is to -nd chaotic,smooth f’s in X from chaotic, compact #-invariant sets of F�(N;X), is discussed inRemark 3.

As a corollary of Theorem 7 we can state the chaoticity of # in F�(N; C1). Whenwe further restrict the domain of # to the (compact and invariant) stationary set H we-nd that sensitivity to initial conditions and density of periodic points are preserved,while transitivity is lost.

Theorem 8. The shift map has in H sensitivity to initial conditions; density of periodicpoints; but it is not transitive.

Proof.. The density of periodic points is due to the density of Hrat in H . To demonstrateSIC, let c�(·) ∈H be given. By density of Krat in K there exists �(·)

r ∈Kr such that,|�(·) − �(·)

r |� ≤ �. Let ’ be the diDerence of the phases of � and �r and let us suppose,by possibly changing �r , that ’ is irrational. The phase diDerence between �n and �n

ris n’, so that there exists an n∈N such that cos(n’)¡ 1

2 . Thus

|#n(c�(·)) − #n(c�(·)r )|� ≥ �|�n − �n

r | = �√

2(1 − cos(n’)) ≥ �

which implies SIC.


Non-transitivity of H is obvious, because it is the union of the family of subsetsH� = {c�(·); |c| = 1} each of which is compact, and invariant.

The complex behavior of the shift map in the whole space F�(N;X) can be furtherunderlined by the analysis of the stable sets.

Referring again to a generic map f in a metric space X , the stable set Sx of x∈Xis de-ned as the collection of all y∈X such that limn→∞ f(n)(y)=x. The =nite stableset S∗

x (⊂ Sx) is de-ned as the collection of all z ∈X such that f(n)(z) = x for somen∈N.

Theorem 9. For any f∈F�(N;X); S∗f is dense in l∞.

Proof.. Let g∈ l∞. Let us consider gn = {Sn(g); f}. Clearly gn ∈ S∗f . From

|gn − g| ≤ supi∈N

|g(n + i) − f(i)|�i+n ≤ (|g|1 + |f|�)�n

it follows that limn→∞ gn = g.

The theory developed so far concerning the dynamical properties of the shift map inthe ∞-dimensional space F�(N;X) demonstrates that such a map provides an exampleof chaotic linear map.

We wish to emphasize the basic role played by the in-nite-dimensionality of thespace in the chaoticity of # by noting that # is not chaotic in F1(N; B) (B compact),i.e. with respect to the sup-norm.

This point can be further discussed from a diDerent point of view. Some of thepreviously established properties of # can be reproduced also in -nite dimension. Forexample, in Cn the density of the periodic orbits can be obtained by the linear maprepresented, in the canonical base, by the matrix of cyclic permutation

0 1 0 0 : : : 00 0 1 0 : : : 00 0 0 1 : : : 00 0 0 0 : : : 0: : : : : : : : : : : : : : : : : :1 0 0 0 : : : 0

:

Actually, this map has only periodic orbits. On the other hand, the property of densityof S∗

0 is exhibited by the nihilpotent matrix

0 1 0 0 : : : 00 0 1 0 : : : 00 0 0 1 : : : 00 0 0 0 : : : 0: : : : : : : : : : : : : : : : : :0 0 0 0 : : : 0


as the origin in Cn is globally attracting for it. These diDerent properties are obtainedby varying the last row of the matrix, while the -rst n− 1 are kept -xed.

When passing to the in-nite dimension, the last row of the matrix is “lost”, andthe cyclic matrix and the nihilpotent matrix merge into the matrix of the shift map. Inother words, the in-nite dimension allows for the same linear map to have propertiesthat in -nite dimension can only pertain to diDerent maps.

5. Chaotic linear maps in function spaces

The shift map in F�(N;C) can be used to obtain examples of chaotic linear maps inspaces of analytic functions. For this, it is enough to -nd an isomorphism ’ :F�(N;C) →E which applies F�(N;C) unto a Banach space of analytic functions, say E; and thento consider the (linear) operator ’ ◦ # ◦ ’−1 :E → E conjugate to the shift map. Letus give some examples.

1. Let us call ’1 the correspondence that associates with each function f∈F�(N;C)the power series in C’1(f)(z) =

∑∞n=0 f(n + 1)zn. For |z| ≤ �¡�;

|’1(f)(z)| ≤ 1z

∞∑n=0

|f(n + 1)|�n+1(��

)n≤ |f|�

�− �

which implies that ’1(f)(z) is analytic in the disk D� = {z ∈C: |z| ≤ �}: ’1 is anisomorphism between F�(N;C) and E1 = ’1(F�(N;C)), when the latterspace is endowed with the Banach-space structure induced by ’1 (in particular|’1(f)|� = |f|�).

Let us identify the linear operator conjugate to # via ’1. Given h(z) =∑∞

n=0 f(n +1)zn ∈E1 we -nd

’1 ◦ #1 ◦ ’−11 (h(z)) =

∞∑n=0

f(n + 2)zn

=∑∞

n=0 f(n + 1)zn − f(1)z

=h(z) − h(0)

z;

namely, ’1 ◦#1 ◦’−11 is the incremental ratio at the origin. Thus, the incremental ratio

is a linear, chaotic operator in E1. (We remark that here chaotic means, has explainedin the previous paragraph, that the map is chaotic in some compact subset of E1 andthat it has in E1 the complex behavior discussed above).

The eigenvectors of the incremental ratio have a good interpretation. In fact, as theeigenvector of the shift map corresponding to �∈C; |�|�¡ 1; is of the form �(·), theeigenfunctions of the incremental ratio are h(z) =

∑∞n=0 �

nzn = 1=(1 − �z).The -xed point ’1(1(·)) is 1=(1−z) and, more generally, the n-periodic points of the

incremental ratio, solutions of h(z)−∑n−1k=0 h(k)(0)zk=zh(z); are of the form P(z)=(1−z),

where P(z) is a polynomial in z of degree n.


2. Let us call ’2 the correspondence that associates with each function f∈F�(N;C)the power series in C’2(f)(z) =

∑∞n=0 f(n + 1)=n! zn. For any z

|’2(f)(z)| ≤ 1�

∞∑n=0

|f(n + 1)|�n+1

n!

(z�

)n≤ |f|�

�e|z=�|

which implies that ’2(f)(z) is analytic in C: ’2 is an isomorphism between F�(N;C)and E2 =’2(F�(N;C)), when the latter space is endowed with the Banach-space struc-ture induced by ’2.

Let us identify the linear operator conjugate to # via ’2: h(z) = ’2(f)(z) can bederived term by term, and the derivative is given by

dh(z)dz

=∞∑n=1

f(n + 1)zn−1

(n− 1)!=

∞∑n=0

f(n + 2)zn

n!: (2)

Comparing (2) with the de-nition of h(z) yields the conclusion that the linear operatorconjugate to the shift map is the derivative. Thus the derivative is a chaotic operatorin E2.

The eigenvectors of the derivative are given by∑∞

n=0 �nzn=n! = e�z, while the -xed

point ’2(1(·)) is the exponential function ez. More generally, the n-periodic points,solutions of dnf(z)=dzn = f(z); are of the form P(z)e�z, where P(z) is a polynomialin z of degree n.

6. Chaotic invertible linear maps

In this section, we extend the analysis of the shift map to suitable Banach spacesof bi-in-nite sequences. The main diDerence with respect to the previous case is thatthe shift map is now invertible, and the inverse has the same properties of #. Thesame basic de-nitions and notations of the foregoing paragraphs apply, except for thesubstitution of N with Z, the set of integer numbers.

In the space F(Z;X), that is the the set of functions whose ranges are bi-in-nitesequences of elements of X, we consider, for 0¡�¡ 1, the norm

|f|� = supn∈Z

{|f(n)|�|n|}: (3)

Theorem 1 can be extended to show that with the above norm the subset of

F�(Z;X) = {f: |f|� ¡ + ∞}is a non-separable Banach space. Also the analogous of Theorem 2 holds true, that isfor any compact subset B of X; F(Z; B) is a compact subset of F�(Z;X).

For the shift map # :F�(Z;X) → F�(Z;X), de-ned as #(f)(n) = f(n + 1); ∀n∈Z,the inverse of # is unique and given by #−1(f)(n) = f(n− 1); ∀n∈Z.

Norm (3) is symmetric with respect to the transformation n → −n. This implies that# and #−1 have the same asymptotic behavior. Non-symmetric norms, such as

|a|�;� = sup

{supn≥0

{|an|�n}; supn¡0

{|an|�−n}}


would imply diDerent dynamics for # and #−1, but this issue will not be discussedhere.

Theorem 3, except for point (v), continues to hold. In particular, the norm of # is�−1 and obviously that of #−1 is �.

0 is no longer an eigenvalue but belongs to the resolvent set. In fact, the spectrumis substantially changed. The point-wise spectrum of # in F�(Z;X) is the annulus� ≤ |�| ≤ �−1. In fact, the solutions of #(f) = �f, are �(·)x; x∈X, but �(·)x belongsto F�(Z;X) if and only if � ≤ |�| ≤ �−1: As in the non-invertible case the essentialspectrum is empty.

The explicit formula for the solution f of the equation #(f)−�f=g when � belongsto the resolvent set is f(n)=

∑∞i=0 �

ig(n−1−i), if |�|¡�, and f(n)=−∑∞i=0 g(n+i)=

�i+1 if �−1 ¡ |�|¡∞.Theorem 5 extends, with suitable rede-nition of the conjugating homeomorphism

Of , to Lipschitz-continuous homeomorphisms f of compact subsets of X.Theorem 6 holds unchanged, apart from suitable rede-nitions of the sets H and K .For what concerns the chaotic nature of #, basically we have the same results of

Section 4, that is

Theorem 10. (i) # is chaotic in F�(Z; B); B compact in X: (ii) SIC and density ofperiodic points occur in the stationary subset H; but not transitivity.

As in Section 5, we can give some examples of “concrete” linear chaotic invertibleoperators by conjugating the shift map in F�(Z;X) to simple maps in spaces of analyticfunctions.

1. With f∈F�(Z;C) let us associate the pair of functions

h1(z) =∞∑n=0

f(n)zn; h2(z) =∞∑n=1

f(−n)zn:

Both functions belong to the space E1. We de-ne the Banach space E3 of pairs offunctions in E1; h(z) = (h1(z); h2(z)); and the map ’3(f) = h: ’3 conjugates # to thefollowing map in E3:

’3 ◦ # ◦ ’−13 (h(z)) =

(h1(z) − h1(0)

z; z(h2(z) + h1(0))

):

The eigenvectors of ’3 ◦ # ◦’−13 can be expressed as h(z) = (1=(1− �z); z=(�− z)).

The -xed point ’3(1(·)) is (1=(1 − z); z=(1 − z)).2. With f∈F�(Z;C) we associate the pair of functions

h1(z) =∞∑n=0

anzn

n!; h2(z) =

∞∑n=1

a−nzn

n!:

Both functions belong to space E2. We de-ne the Banach space E4 of pairs of functionsin E2 h(z) = (h1(z); h2(z)) and the map ’4(f) = h: ’4 conjugates # to the following


map in E4:

’4 ◦ # ◦ ’−14 (h(z)) =

( ∞∑n=0

an+1zn

n!;

∞∑n=1

a−n+1zn

n!

): (4)

Let us now observe that dh1(z)=dz =∑∞

n=0(an+1zn)=n! and that∫ z

0 h2(z) dz =∑∞n=1(a−n+1zn)=n! − zh1(0), so that (4) can be given the explicit form

’4 ◦ # ◦ ’−14 (h(z)) =

(dh1(z)

dz; zh1(0) +

∫ z

0h2(z) dz

):

The eigenvectors of ’4 ◦ # ◦ ’−14 are (e�z; ez=� − 1). The -xed point ’4(1(·)) is

(ez ; ez − 1).

References

[1] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Denjamin=Cummings, Menlo Park, CA,1986.

[2] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of VectorFields, Springer, New York, 1983.

[3] L.V. Kantorovich, G.P. Akilov, Analisi Funzionale, Mir, Mosca, 1980.[4] A.N. Kolmogorov, S.V. Fomin, Elementi di Teoria elle Funzioni e di Analisi Funzionale, Mir, Mosca,

1980.[5] C. Mira, Chaotic Dynamics, World Scienti-c, Singapore, 1987.[6] S. Wiggins, Global Bifurcation and Chaos – Analytical Methods, Springer, New York, 1988.[7] S. Wiggins, Introduction Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990.

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Chaotic linear maps